Question:

AB is a diameter of a circle centred at the origin O, and P is a point on the circumference of the circle. By considering the position vectors of A, B and P, prove that AP is perpendicular to BP

Notifications

Clear all

Dec 09, 2019 5:27 pm

Question:

AB is a diameter of a circle centred at the origin O, and P is a point on the circumference of the circle. By considering the position vectors of A, B and P, prove that AP is perpendicular to BP

2 Replies

Dec 09, 2019 6:28 pm

If two direction vectors have a dot product of 0, they are perpendicular to each other.

You want to prove AP is perpendicular to BP, lets define AP as P-A and BP as P-B.

Now lets find the dot product of them and it should equal zero after some simplifying.

AP.BP = (p-a).(p-b)

Expanding the right hand side we get

p^2 - b.p - a.p + a.b

Now from the diagram:

You can see that b = -a, as they make the diameter. So if we include that in our expression we get

p^2 + a.p - a.p + a.b

so the a.p cancels out and we're left with

p^2 + a.b

And again we're going to plug in the b = -a

p^2 - a^2

Now p^2 actually just means p.p, and if you dot product a vector by itself, you actually have the square of it's magnitude, i.e. |p|^2

so p^2 - a^2 = |p|^2 - |a|^

And hold on a second.... They are all just points on a circle right? So the modulus of p and a must be equal!

|p|^2 - |a|^2 = 0

There are bits of this proof you may not understand as a lot of it is mainly in the old spec - its not homework from school or anything is it?

As you know, the topics for GCSE and A-Level exams have been released today by all exam boards. Below are...

View Post
Choosing the right university is one of the most important decisions you will make in your life. For most students,...

View Post
Ofqual and the government have announced that they are in talks about the changes for the 2022 examinations (GCSE, AS-Level...

View Post
A physics degree will be a good choice for you if you enjoy studying A-Level Maths and Physics. The degree...

View Post
Book a free trial before purchasing, fill in the form below.